This preview shows pages 1–2. Sign up to view the full content.
1
Introduction to
Microeconomics
Lecture 3
Maths: Algebra
Simon Cowan
Outline
z
Solving equations
z
Linear and quadratic equations
z
Solving simultaneous equations
z
Linear demand and supply equations
z
Indices and logarithms
Solving equations: an example
Let’s suppose that a firm is the only supplier of operating
systems and the demand for operating systems is given by
q
=
A
−
Bp
where
q
is the number of units bought,
p
is the price
and
A
and
B
are positive but unspecified numbers. This
holds whenever
0
≤
p
≤
A
/
B
.
(Why?)
If the quantity bought is
10
, what must be the price?
This asks us to find
p*
in
10 =
A
Bp*.
Take
Bp*
to the lefthand side and subtract
10
from both sides to
give
Bp* = A
10
.
Thus
p
* = (
A
10)/
B
is the solution.
We need to check that this satisfies the constraint on the price.
(i)
It is certainly true that
p
* <
A
/
B
because
p
* =
A
/
B
10/
B
.
(ii)
For
p
*
≥
0
we require that
A
≥
10
.
A quadratic equation (i)
Now the firm is threatened with a competition case
for abusing its dominant position.
Suppose that there is a maximum level of profit
that the competition authority will allow,
denoted by
M
Assume this is nonnegative
denoted by
.
Assume this is non negative
(otherwise the firm goes bankrupt).
What are the feasible prices that the firm can set to
ensure that profits equal
M
?
Suppose the firm’s costs consist entirely of
Research and Development (R&D) costs which
are fixed and thus independent of output.
Denote the fixed cost by
F
> 0.
Quadratic equation (ii)
The firm’s profit is
Revenue minus Cost
= Price x Quantity
−
Cost
= p
(
A
Bp
)
−
F
The feasible prices must satisfy
p
(
A
Bp
)
−
F = M
Rearranging yields an
equation
which is quadratic
in
p
:
Bp
2
−
Ap + F + M =
0.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 04/12/2010 for the course ECON DEAM taught by Professor Vines during the Spring '10 term at Oxford University.
 Spring '10
 Vines
 Economics, Microeconomics

Click to edit the document details